• defining differential operator (as gradient and divergence) on manifold
• understand how Laplacian is defined thanks to Laplace-Beltrami operator
• understand PDEs on manifold and define the Brownian motion on it
• how to construct the Brownian motion on Riemanian manifold with the Laplace-Beltrami operator
• several concrete examples

• Newtonian Classical Mechanics (affine spaces, Galilean transformations, Newton's equation)
• Newtonian Classical Mechanics (examples, kinetic and potential energy)
• Lagrangian Classical Mechanics (motivation, calculus of variations)
• Lagrangian Classical Mechanics (Lagrangian function and action, Euler-Lagrange formula, Hamilton's equation)
• Observables (algebra of observables, Poisson-bracket, analogies to quantum mechanics)
• Axioms of Quantum Mechanics (states, observables, measurement, correspondence principle, position and momentum operators)
• Schrödinger Equation (wavefunction, Hamilton operator, eigenvalue equation and eigenstates)
• Schrödinger Equation continued (eigenstates for potential well and potential step), Double-Slit Experiment (interference, probability amplitude calculus)

• Tangent Vectors, The Differential of a Smooth Map, The Tangent Bundle (reference: “Introduction to Smooth Manifolds” by John M. Lee)
• Connection, Vector Fields, Covariant Derivative, Geodesic and Parallel Translation
• Discussion about curvature of space and Gauss-Bonnet Theorem
• Divergence theorem, Green's identities
• Laplace–Beltrami operator
• Lie group and principal bundle

• differentiable structures, manifolds, tangent spaces
• de Rham cohomology, embeddings of manifolds, regular value theorem
• (pseudo-)Riemannian manifolds, Riemannian measures, Laplace-Beltrami operator, Gauss-Green identity, Stokes' theorem
• Covariant derivative, Riemannian connection, Christoffel symbols, Parallel displacement, exponential mapping (cf. Section 5D, Kuehnel: Differential geometry)
• Curvature tensor and intrinsic geometry of surfaces: Equations of Gauss and Weingarten, Theorema egregium, Fundamental theorem of the local theory of surfaces (Sections 4A-D, Kuehnel)
• More on the curvature tensor and Weingarten map, Sectional curvature, Ricci tensor and curvature, Einstein tensor (Chapters 4,6: Kuehnel)
• Spaces of constant curvature: Hyperbolic space, Geodesics and Jacobi fields, Local isometry of spaces of constant curvature (Chapter 7: Kuehnel)

• Organizational meeting
• Probability that random polynomial has no real roots (Talk by Anna Gusakova)
• Theorems on positive real functions (Talk by Anna Muranova)
• Non-explosion of solutions to SDEs with singular coefficients (Talk by Chengcheng Ling)
• Emergence of Flocking in the Fractional Cucker-Smale Model (Talk by Peter Kuchling)
• Robust Poincaré inequality and Application to the transitional phase of fractional Laplacian (Talk by Guy Fabrice Foghem Gounoue)
• My problems and failed ideas regarding the edge fluctuation of non hermitian random matrices with independent entries (Jonas Jalowy)
• Effective impedance of a finite electric network. Definitions and main properties (theorems and conjectures) (Anna Muranova)
• Application of Probability Methods in Number Theory and Integral Geometry (Anna Gusakova)
• Ordered fields. The ordered field of rational functions“ (Anna Muranova)
• Dynamics on the cone: an overview of my thesis (Peter Kuchling)
• Well-posedness of SDE driven by Levy noise (Chengcheng Ling)
• Discussion about the talks for workshop/retreat

• Point processes 3: Moment Problem, Local Convergence Jansen, Gibbsian Point Processes, pp. 33 - 45
• Point processes 2: Intensity Measure, Correlation Functions, Generating Functionals Jansen, Gibbsian Point Processes, pp. 25 - 33
• Point processes 1: Configuration Spaces, Observables, PPP Jansen, Gibbsian Point Processes, pp. 15 - 25
• Introduction to partition functions
• Partition Functions and Gibbs Measures
• Correlation Functions
• Phase Transitions (Critical Exponents and Models)
• Scaling Limits
• Mean Field Limit of the Cucker-Smale model: Two Approaches (Peter Kuchling)

Reading Spectral graph theory by Fan R. K. Chung

• Chapter 2: Isoperimetric problems
• Chapter 3: Diameters and eigenvalues
• Talk by Anna Muranova: On effective resistance of electric network with impedances
• Talk by Melissa Meinert: Ollivier Ricci curvature for general Laplacians
• Markov chains
• Talk by Filip Bosnic: Examples of Poincaré and log-Sobolev inequalities on discrete configuration spaces
• Introduction to random graphs by Alan Frieze and Michał Karoński

Discuss Lévy Processes and Infinitely Divisible Distributions by Ken-iti Sato

• Chapter 1 (pp. 1 - 30): Examples of Levy Processes
  
• Chapter 2 (pp. 31 - 68): Characterisation and Existence of Levy Processes and Additive Processes
• Chapter 2 (pp. 31 - 41): Infinitely Divisible Processes and the Lévy-Khintchine formula
• Chapter 2 (pp. 54 - 67): Transition Functions and the Markov Property; Existence of Lévy and Additive Processes
• Chapter 3 (pp. 69 - 77): Selfsimilar and Semi-selfsimilar Processes and their Exponents
• Chapter 3 (pp. 77 - 90): Representations of Stable and Semi-stable Distributions
• Viswanathan et al. Papers: “Levy Flight Search Patterns of Wandering Albatrosses” and “Optimizing the Success of Random Searches”
• Bertoin (pp. 18-24): Potential theory: Markov property and important definitions
• Bertoin (pp. 43-48): Duality and time reversal
• Bertoin (pp. 48-56): Potential theory: capacity measure; essentially polar sets and capacity
• Bertoin (pp. 56-68): Potential theory: energy; the case of a single point
• Bertoin (pp. 71-84): Subordinators: definitions; passage across a level; arcsine laws
• Bertoin (pp. 84-99): Subordinators: rates of growth; (Hausdorff-)dimension of the range

Discuss

• Maximum principle for ecliptic operators (Gibarg-Trudinger)
• Paley-Littlewood Decomposition (Grafakos classical Fourrier Analysis, second or third edtion)
• Another looks at Sobolev spaces (Articles Brezis Bourgain_Mironescu)
• Spectral theory: spectral measure for Unbounded operators
• Spectral theory for unbounded operator and spectral measure (Book by Reed-Simon)
• Restriction of the Fourier Transform on the sphere (Book by Elias Stein)
• Interplation of Banach spaces and application (Lecture note by Alessandra Lunadri Theory of interpolation 2009)
• Introduction to optimal transport theory (Book by Cedri Villani: Old and New, it is free on-line)
• Introduction to control and non linearity (Book by Jean-Michel Coron: Control and Non-linearity free one his we-page)
• Introduction to Calculus of variations